论文标题
高斯聚合物模型中接地状态的几何形状
The geometry of near ground states in Gaussian polymer models
论文作者
论文摘要
最大化路径的能量和几何形状在可集成的最后一段渗透模型中受到三分之一和三分之二的特征性kpz缩放指数的控制。当在尊重这些指数的尺度坐标中表示时，该路径的随机场可能被视为复杂的能量景观。我们研究了该景观中山谷的结构和连接途径。路由权重配置文件$ \ mathbb {r} \ to \ mathbb {r} $ aSsociates to $ x \ in \ in \ mathbb {r} $通过一条路径可获得的最大缩放能量，该路径可从$（0,0）$（0,0）$（0,0）$（0,1）$通过点$（x，x，1/2）$获得。从[HAM16]和[CHH19]开发了Brownian Gibbs分析的工具，我们证明了该概况的强烈相似性，即Brownian的最后一次通过对单位级尺度上的速率二次运动的最后一段渗透。对稀有性的尖锐估计值是，在能量景观中两个宏观不同的途径提供了接近全球最大结果的能量。我们证明了关于最大化路径的能量和几何形状的连续性模量的强大断言，这些模量发展了[HS20]的结果和方法，从而在显微镜上方的所有尺度上提供了有效的估计值。研究了关于最大化路径的近距离偏移的几何形状：的确，我们估计了缩放路径的能量短缺，被迫紧密地模仿最大途径的几何形状，同时保持与该路线的不一致。我们还提供最大化路径的近似梯度的边界，被视为功能，排除了急剧的陡峭运动至微观尺度。我们的结果在动态扰动下的稳定性和脆弱性的同伴研究[GH20A]中发现了应用。
The energy and geometry of maximizing paths in integrable last passage percolation models are governed by the characteristic KPZ scaling exponents of one-third and two-thirds. When represented in scaled coordinates that respect these exponents, this random field of paths may be viewed as a complex energy landscape. We investigate the structure of valleys and connecting pathways in this landscape. The routed weight profile $\mathbb{R} \to \mathbb{R}$ associates to $x \in \mathbb{R}$ the maximum scaled energy obtainable by a path whose scaled journey from $(0,0)$ to $(0,1)$ passes through the point $(x,1/2)$. Developing tools of Brownian Gibbs analysis from [Ham16] and [CHH19], we prove an assertion of strong similarity of this profile for Brownian last passage percolation to Brownian motion of rate two on the unit-order scale. A sharp estimate on the rarity that two macroscopically different routes in the energy landscape offer energies close to the global maximum results. We prove robust assertions concerning modulus of continuity for the energy and geometry of scaled maximizing paths, that develop the results and approach of [HS20], delivering estimates valid on all scales above the microscopic. The geometry of excursions of near ground states about the maximizing path is investigated: indeed, we estimate the energetic shortfall of scaled paths forced to closely mimic the geometry of the maximizing route while remaining disjoint from it. We also provide bounds on the approximate gradient of the maximizing path, viewed as a function, ruling out sharp steep movement down to the microscopic scale. Our results find application in a companion study [GH20a] of the stability, and fragility, of last passage percolation under a dynamical perturbation.
